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Mathematics versus Engineering

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Introduction

The first engineer known to history was the Egyptian legend Imhotep, circa 2550BC, who created one of the most astounding engineering marvels of all time, the Great Pyramids at Gizeh. Besides being a magnificent sight, the pyramids are a miracle of mathematics: the surface area of each face of the pyramid equals the square of its height; whether inside or outside, all its dimensions, orientations and angles are perfect; massive granite blocks are cut precise to one tenths of a millimetre; the slopes of the pyramid diverge in perfect proportion from the apex, in a magnificent convex that seems to embrace the sun. This is but one historical instance of millions, which shows engineering ingenuity through applied mathematics.
All engineering can be arguably said to rely heavily on mathematic fundamentals. Through the study of mathematical quantities and relations, an engineer can devise a system to exacting requirements and attain new heights in his practice.
This assignment aims to throw necessary light on how a compendium of many mathematical branches serves as necessary tools for engineering.

Task 1
A lathe is a machine tool which spins a block of material to perform various operations such as cutting, sanding, knurling, drilling, or deformation with tools that are applied to the work piece to create an object which has symmetry about an axis of rotation.

a) Here, the lathe tool has to accommodate work between two set extremities. For convenience and commonality of solution, we include these extremities as part of our calculations.
Bar diameters D1, D6 = 25mm, 300mm respectively.
Cutting speed S = 25mm min-1 = 25000mm min-1
No of spindle speeds n = 6

As , =318.31rpm
Similarly, =26.53 rpm

i) An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
If a1 is the first term, an is the nth term and d is common difference, then
Here,
That is, = —58.36 rpm
The sequence of spindle speeds is hence:
318.31, 259.95, 201.59, 143.23, 84.87, 26.51 in rpm.
Bar diameters can be determined using as follows:
25, 30.61, 39.47, 55.56, 93.76, 300 in mm.

ii) A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number.
If a1 is the first term, an is the nth term and r is common ratio, then
Here, =0.

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6083
The sequence of spindle speeds is hence:
318.31, 193.63, 117.78, 71.64, 43.58, 26.51 in rpm.

Bar diameters can be determined using as follows:
25, 41.09, 67.56, 111.07, 182.6, 300 in mm.

b) AP graph

GP graph

c) AP graph

GP graph

Among spindle speeds in AP and GP, GP is preferred, as it gives a closer fit to ideal cutting speed over all possible sizes, although only six sizes can be cut at exactly 25m min-1.

d) Let the length of the first piece
x cm. By the definition in solution a) i),
the lengths of the other two pieces will be
xr cm and xr2 cm respectively, where r is the
common ratio.
By the question,
We also have
Hence
Or,

Or,
Or,
This is a quadratic equation which can be reduced to .

Sometimes, it is useful to know the characteristics of the equation by plotting its graph:



As r is a multiple of length, it cannot be negative; hence the second solution is rejected.
 r = 2  =7cm.
Therefore, the lengths of each piece are 7, 14 (72) and 28 (722) cm.
Task 2
a) We can plot graphs with the help of comprehensive tables:


(Degrees)
(Units)

-360 0
-300 26
-240 26
-180 0
-120 -26
-60 -26
0 0
60 26
120 26
180 0
240 -26
300 -26
360 0



(Degrees)
(Units)

-360 -25
-300 25
-240 50
-180 25
-120 -25
-60 -50
0 -25
60 25
120 50
180 25
240 -25
300 -50
360 -25



(Degrees)
(Units)

-360 -25
-300 51
-240 76
-180 25
-120 -51
-60 -76
0 -25
60 51
120 76
180 25
240 -51
300 -76
360 -25

It is useful to plot a comparison graph between :


In order to express the function in terms of , we observe the graph:

Clearly,
A = 77.5units

To determine β, we compare the graph (green) to a cosine curve (blue).
The comparison plot is as follows:


Clearly, β = -108o
Hence,
b)




Graph of i vs t :

i is maximum at 5 A. This occurs when sin x = 1 or at x =
Here,

1.4745C
i is maximum at this point of time.

c)

This cannot be solved algebraically.
Using higher methods, we get x=.63488o , or no solution.

Task 3
a) A real power series is a series of the form , where are real numbers, and is a real variable.
By identity,
Theoretically,

Expanding for gives
Or,
This can be otherwise stated as
For we substitute for as follows:
=1.128 correct to 3d.p.
A graphical interpretation is useful, as given:

b)
This can be expressed as
Or otherwise,
We can now substitute values for as stated:
At = —2 , B = —1
At = 0 , A+B+D = 0; or otherwise, A+D = 1
At = 1 , 2(A+D) + B + 2C = 1; or otherwise, C = 0
At = 2 , 6A + 5B + 18C + 9(A+D) = 4; or otherwise A = 0
From the second statement, D = 1
Hence, we get


is an asymptote.

c) A graphical representation of the form is:

It is useful to determine a straight-line graph to confirm the adherence of results to the law, given by form, that is:

This serves as a graphical confirmation for the relation.
Therefore, constant parameters can be determined by solving any two equation sets, given by

This gives and .

d) We are given
Simplifying,

Differentiating both sides w.r.t. ,
To find the value of , we equate to 0,


This implies gives ( is a turning point)
= 9

Task 4
a) We are given speed of sound in a gas is given by
As , we can get
We make the following assumptions
At velocity , gas temperature is related to velocity as
At velocity , gas temperature is related to velocity as
It is observed that and are constants, hence
, and

Taking ratios, we get
Or otherwise,
b) Van der Waals equation of state for non-ideal fluids is given by
We can re-arrange the equation as
We are given the following facts and
In other words, a graph has a turning point.
Differentiating both sides w.r.t. , we get (A)
Again differentiating, we get (B)
Hence we get a set of simultaneous equations.
In order to solve them, we multiply the first eqn. by to yield
(C)
Adding (B) and (C), we get an equation devoid of terms with :

Or otherwise, to give
Substituting this value in (A) yields
Simplifying, we get
Or otherwise,

c) An equation to model logistic growth can be stated as
In all applications, exponent of is negative for all positive .
Hence, as ,
(1)
(2)
(3)
Thus is the upper limit of in this growth model. It has a special name called carrying capacity.
Thus the function can be categorised as a sigmoid, that is, a real-valued and differentiable, having either a non-negative or non-positive first derivative and exactly one inflection point.

The graph of such a function is in an S shape. Functional parameters can also be defined on graphical basis.

A sigmoid curve looks like this:

The given restricted-growth function is
i)
ii)
Setting =100 and ,

iii)
gives amount of at an instant, that is, instantaneous growth rate.

It is useful to observe the graphs of and

The curve of the derivative is a classic Hubbert’s curve or regular bell curve.
d) We are given
Substituting for gives
Or otherwise,
Solving, we get —4, —2 from

Overview
From calculations of heat and mass flow in steam power plants, to calculations of molten metal flowing and mixing in weld pools, applied mathematics turns the wheels of engineering analysis and design.

What thickness of concrete is sufficient for the roof of a 3 m wide tunnel? How many tons of air conditioning are needed to maintain a temperature of 20°C in an office of 200 square meters floor area, 42 people, 35 personal computers, 26 windows and 6 doors? How much rocket fuel is needed to transport one kilogram of gold to the moon?

These are technological questions, as opposed to scientific or mathematical questions. These are some problems faced by engineers. They neither seek fundamental truth about nature, nor require some pure standard of intellectual rigour in their answers. They also supply incomplete information and leave room for the engineer to make simplifying assumptions, develop models, carry out calculations, draw on prior experience, and use safety factors to obtain reasonable answers with reasonable effort.

The process of training an engineer to answer such questions requires a study of engineering models and more importantly, the mathematical techniques used to analyse them. These mathematical techniques require practice, sophistication and rigour. In this way, the technological world of an engineer builds up from mathematical discipline.

Yet, engineering is exciting and worthwhile. It is the name for activity geared towards the purposeful exploitation of the laws, forces and resources of nature, not merely towards uncovering further esoteric truths but towards a direct improvement of the human condition. Mathematics serves as a can – opener, opening new vistas of scientific knowledge and mental activity.



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